Determining non-permissible values for trig expressions

Determining Non-Permissible Values in Trig

When dealing with trigonometric functions and expressions, oftentimes we will encounter and be asked to solve for trig values for which an expression is “non-permissible” – that is, our answer would be undefined. This most often occurs when trig values equal zero, for example sin0. These values are also called non-permissible values when they result in an expression that is undefined. Below is a trig values chart that has the exact values of trig functions for sine, cosine, and tangent.

Value chart of sine,
cosine, tangent function

NOTE: This chart just gives the values for sine, cosine, and tangent in the first quadrant using the common reference angle. If you recall, these values will vary in their sign (+/-) depending on which quadrant the angle is in. We can use the acronym ASTC (All Students Take Calculus) to help us to remember which trig ratio is what in each quadrant:

Pythagorean Identities:

Though this should be second nature to you now, it is important to make sure you have a firm grasp of these most basic identities.

In order to understand trigonometric ratios, let’s first look back on some basic trigonometry principles. First, the Pythagorean Theorem with right triangles, and SOHCAHTOA. If you recall, we can use the Pythagorean Theorem (a2+b2=c2)(a^{2} + b^{2} = c^{2})(a​2​​+b​2​​=c​2​​) to solve for unknown side lengths of right triangles, and we can use SOHCAHTOA to find missing angles. Below are the formulas we get from SOHCAHTOA, as well as an image to help you visualize it:

Quotient Identities:

In trigonometry, quotient identities refer to trig identities that are divided by each other. There are two quotient identities that are crucial for solving problems dealing with trigs, those being for tangent and cotangent. Cotangent, if you’re unfamiliar with it, is the inverse or reciprocal identity of tangent. This identity will be more clear in the next section. Below, this image covers the two fundamental identities you must know when it comes to quotient identities.

Quotient identities

Reciprocal Identities:

Have you ever wondered if there was an easier way of dealing with the trigonometric expressions such as sin−1x\sin^{-1} xsin​−1​​x? It turns out, there is. In trigonometry, reciprocal identities or inverse identities cover this base. Instead of writing sin−1x\sin^{-1} xsin​−1​​x or 1sinx\frac{1}{\sin x}​sinx​​1​​ , we can use the reciprocal identity cscx\csc xcscx instead. Cosecant (csc), secant (sec), and cotangent (cot) are extremely useful identities, and you will use them extensively as you progress with mathematics into pre-calculus and calculus. Therefore, it is quintessential that you memorize and understand all of these identities. The image below covers what you must know.

Reciprocal identities

How to Find Point Discontinuity:

Now that we have gone over the most important identities used to simplify trig expressions, we can focus in on how to find points of discontinuity. Of course, the best way to learn this is to do a couple of example problems:

Example 1:

Find the point(s) of discontinuity for the following trig expression:

cosx1+2sinx\frac{\cos x}{1+2\sin x}​1+2sinx​​cosx​​

Step 1: Find the Expression of Discontinuity

As mentioned earlier, non-permissible values occur when an expression is undefined, most often when the denominator equals zero. In this case, let’s make the denominator equal to zero and simplify.

1+2sinx≠01 + 2\sin x \neq 01+2sinx≠0

sinx≠−12\sin x \neq -\frac{1}{2}sinx≠−​2​​1​​

Step 2: Solve for Values of x

Now we need to solve for x to find which values of the variable result in sinx being equal to −12-\frac{1}{2}−​2​​1​​. According to ASTC, sine is negative in the third and fourth quadrant.

Determine the sign using ASTC

Next, because we know that sin30\sin 30sin30 (or sinπ6\sin \frac{\pi}{6}sin​6​​π​​) is equal to 12\frac{1}{2}​2​​1​​, all we need to do is use this reference angle in each of these quadrants.

Using reference angle in third quadrant

Using reference angle in the forth quadrant

This leaves us with our answers, which are our points of discontinuity:

Again, as mentioned before, we often need to use trig identities to simplify our expressions before we go solving for non-permissible values. In this case, we use reciprocal trig identities to simplify and find the expression for the points of discontinuity. We will find two expressions for this in this particular example.

1cosx1−cosx\frac{\frac{1}{\cos x}}{1 - \cos x}​1−cosx​​​cosx​​1​​​​

cosx≠0,1−cosx≠0\cos x \neq 0, 1 - \cos x \neq 0cosx≠0,1−cosx≠0

Simplification using trigonometric identities

Step 2: Solve for Values of x

Again, now we need to solve for x to find which values of the variable result in cosx\cos xcosx being equal to 0 or 1. In this case, we don’t need to use ASTC because we can easily find these values through the graph of cosine.

cosx≠0\cos x \neq 0cosx≠0

x≠π2,3π2x \neq \frac{\pi}{2}, \frac{3\pi}{2}x≠​2​​π​​,​2​​3π​​

1≠cosx1 \neq \cos x1≠cosx

x≠0,2πx \neq 0, 2\pix≠0,2π

This leaves us with our final answers, which are our points of discontinuity: